Video: CBSE Class X • Pack 4 • 2015 • Question 19

CBSE Class X • Pack 4 • 2015 • Question 19

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Video Transcript

A hemisphere is mounted on top of a cube of side 10 centimeters, with its flat face resting on the cube. If the flat face of the hemisphere is completely contained by the adjoining face of the cube, what is the largest diameter that the cube can have? Find the cost of painting the surface of the combined solid, with a hemisphere of maximum diameter, at a rate of five rupees per centimeter squared. Use 𝜋 equals 3.14.

So in this question, we have a hemisphere mounted on top of a cube with the flat face of the hemisphere resting on the cube. So let’s draw a quick sketch of what this would look like. We’re also told that the cube has a side length of 10 centimeters. So we can label this on our diagram.

Another crucial piece of information that the question tells us is that the flat face of the hemisphere is completely contained by the adjoining face of the cube. And what this means is that the flat face of the hemisphere which is touching the cube is contained within the square top of the cube. And what the question is asking is what’s the maximum diameter of this circular face of the hemisphere such that it remains completely contained within the face of the cube.

Now, the largest possible hemisphere which could remain contained within the square face of the cube would have the two widest parts of its circular face touching the edges of the square face of the cube. Now, let’s look at a top-down view of what this would look like.

Now, the circle in this diagram will be identical to the circular face of the hemisphere. And as we can see, the widest part of the circle is touching the edges of the square. And now, as we know, the widest part of the circle is actually its diameter. And since the square is one of the faces of our cube, we know that it has a side length of 10 centimeters. This means that the largest diameter of this circle and therefore of the hemisphere is 10 centimeters.

So now, we have answered the first part of this question. Next, we’re asked to find the cost of painting the surface of this combined solid, where the hemisphere has a maximum diameter. And this is in fact what we’ve just found. Now, in order to find the cost of painting this solid, we first need to find its surface area.

Now, if we’re finding the surface area of the shape, we can look at our diagram to help us out. So the total surface area will be equal to the surface area of the cube minus the area of the circular face of the hemisphere, which is touching the cube, plus the surface area of the curved part of the hemisphere.

Now, let’s calculate each of these parts individually. In order to find the surface area of the cube, we can find the surface area of each individual face since the cube has six identical square faces. So the surface area of one of the cubes’ faces is simply is length squared. So that’s 10 times 10 centimeters, which gives us 100 centimeters squared. And since there are six square faces on a cube, that means that the total surface area of the cube is six times 100 centimeters squared, which is equal to 600 centimeters squared.

Next, we’ll find the area of the flat circular face of the hemisphere. Now, we know that the circle has a maximum possible diameter. And since the radius is equal to half of the diameter, this means that the radius must be equal to 10 over two centimeters or five centimeters.

And now, the area of a circle is given by the formula 𝜋𝑟 squared, where 𝑟 is the radius of the circle. And since the radius of our circle here is five centimeters, this means that its area is 𝜋 timesed by five centimeters squared or 25𝜋 centimeters squared.

Now that we found the radius of our hemisphere, it’ll be easier to calculate the curved surface area of it. And now, the surface area of a sphere is equal to four 𝜋𝑟 squared. And so the surface area of the curved part of the hemisphere will be equal to half of this amount, which is also equal to two 𝜋𝑟 squared.

So in order to find our curved surface area, we simply substitute 𝑟 equals five centimeters into this equation. And this gives us two 𝜋 times five centimeters squared. And this is simply equal to 50𝜋 centimeters squared. So now, we have found the total surface area, we just need to simplify this equation.

We have that this is equal to 600 plus 25𝜋 centimeters squared. And now, we can substitute in the approximation of 𝜋 of 3.14. Multiplying the 3.14 by 25, we obtain 78.5. And for our final step in finding the total surface area of this solid, we simply add the 600 and the 78.5 to leave us with 678.5 centimeters squared.

Now, all that remains to do is to find the cost of the paint to cover this total surface area. So we have that the cost of the paint is five rupees per centimeter squared. So the total cost is equal to the total surface area, which is 678.5, multiplied by the rate, which is five. And since 678.5 multiplied by five is equal to 3392.5, the total cost of painting this combined solid is 3392 rupees and 50 paise.

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